A new randomized vector algorithm for iterative solution of large linear systems

Abstract

In this letter we suggest a new randomized scalable stochastic-matrix-based algorithms for calculation of large matrix iterations. Special focus is on positive or irreducible nonnegative class of matrices. As an application, a new randomized vector algorithm for iterative solution of large linear systems of algebraic equations governed by M-matrices is constructed. The idea behind these stochastic methods is in a randomized vector representation of matrix iterations. The iterations are performed by sampling random columns only, thus avoiding not only matrix but also matrix vector multiplications. As a result, the algorithm is highly efficient for solving linear equations of high dimension, its computational cost depends linearly on the dimension. Extensions of the suggested randomized iteration method to general classes of matrices are also discussed.